Problem 71 - Calculus and its applications

Question

In New York City, taxicabs charge passengers \(\$ 2.50\) for entering a cab and then \(\$ 0.50\) for each one-fifth of a mile (or fraction thereof) traveled. (There are additional charges for slow traffic and idle times, but these are not considered in this problem.) If \(x\) represents the distance traveled in miles, then \(C(x)\) is the cost of the taxi fare, where $$ \begin{array}{ll} C(x)=\$ 2.50, & \text { if } x=0 \\ C(x)=\$ 3.00, & \text { if } 0 < x \leq 0.2 \\ C(x)=\$ 3.50, & \text { if } 0.2 < x \leq 0.4, \\ C(x)=\$ 4.00, & \text { if } 0.4 < x \leq 0.6, \end{array} $$ and so on. The graph of C is shown below. Using the graph of the taxicab fare function, find each of the following limits, if it exists. $$ \lim _{x \rightarrow 0.6^{-}} C(x), \lim _{x \rightarrow 0.6^{+}} C(x), \lim _{x \rightarrow 0.6} C(x) $$

Step-by-Step Solution

Answer

\( \lim_{x \to 0.6^-} C(x) = \$4.00 \), \( \lim_{x \to 0.6^+} C(x) = \$4.50 \), and \( \lim_{x \to 0.6} C(x) \) does not exist.

Step 1: Identify the One-sided Limit as x Approaches 0.6 from the Left

The notation \( \lim_{x \to 0.6^-} C(x) \) asks for the cost as the distance \( x \) approaches 0.6 miles from values less than 0.6. Looking at the given function, for \( 0.4 < x \leq 0.6 \), the cost \( C(x) = \\(4.00 \). Thus, \( \lim_{x \to 0.6^-} C(x) = \\)4.00 \).

Step 2: Identify the One-sided Limit as x Approaches 0.6 from the Right

The notation \( \lim_{x \to 0.6^+} C(x) \) requests the cost as \( x \) exceeds 0.6 miles and comes to 0.6 from values greater than 0.6. For \( 0.6 < x \leq 0.8 \), the cost \( C(x) = \\(4.50 \). Thus, \( \lim_{x \to 0.6^+} C(x) = \\)4.50 \).

Step 3: Determine the Two-sided Limit as x Approaches 0.6

The notation \( \lim_{x \to 0.6} C(x) \) signifies the limit of \( C(x) \) as \( x \) approaches 0.6 from both directions. For a limit to exist, the limits from the left and right must be equal. However, \( \lim_{x \to 0.6^-} C(x) = \\(4.00 \) and \( \lim_{x \to 0.6^+} C(x) = \\)4.50 \) are not equal. Hence, \( \lim_{x \to 0.6} C(x) \) does not exist.

Key Concepts

One-sided limits

One-sided limits in calculus help us understand the behavior of a function as it comes close to a particular point from one specific direction—either from the left or the right.
One-sided limits are significant because they provide a more refined analysis of functions, especially when the function might behave differently approaching from either side of a point.

Let's consider our problem where we need to find the limit of the taxi fare function as it approaches 0.6 miles. There are two one-sided limits to consider:

• Left-hand limit (approaching 0.6 from values lesser): Designated as \( \lim_{x \to 0.6^-} C(x) \) in mathematics notation. As we approach from the left side (or values slightly less than 0.6), our fare is consistent with \(4.00, which makes \( \lim_{x \to 0.6^-} C(x) = 4.00 \).
• Right-hand limit (approaching 0.6 from values greater): Indicated by \( \lim_{x \to 0.6^+} C(x) \). Approaching 0.6 from above (values just beyond 0.6), the fare changes to \)4.50.
  This results in \( \lim_{x \to 0.6^+} C(x) = 4.50 \).

For the taxi fare function, the different costs from left and right at 0.6 arise from the structure of the pricing. By analyzing one-sided limits, we observe how costs are affected at the junctions of fare changes.

Two-sided limits

Two-sided limits consider both directions as a point is approached, which means analyzing the function's behavior from the left and the right simultaneously.
This approach verifies if a function stabilizes to the same value, no matter the direction of approach.

In our exercise, the two-sided limit is expressed as \( \lim_{x \to 0.6} C(x) \) and is dependent on the agreement of the one-sided limits.
Let's recap:

• The left-hand limit as we approach 0.6 is 4.00.
• The right-hand limit as we approach 0.6 is 4.50.

Here’s the key point: For the two-sided limit to exist, both one-sided limits must agree—it means the function should approach to the same value from any side.
In this taxi fare problem, since 4.00 (from the left) and 4.50 (from the right) do not equal, the two-sided limit at 0.6 doesn't exist.
This tells us that there's an abrupt change in the fare, indicating a real-life application of discontinuity in this context.

Piecewise functions

A piecewise function is defined by multiple sub-functions, each applied to a certain interval of the main function's domain.
It looks like a collection of different functions glued together, each governing specific sections based on given conditions.

In the scenario of the taxi fare, we see that the cost function \( C(x) \) is piecewise:

• The base fare is \(2.50, which applies when no distance is traveled (the taxi only enters).
• As distances increase into intervals governed at various lengths (like 0.2, 0.4, 0.6 miles, etc.), the fare also rises in systematic increments, reflecting the piecewise nature.

Each sub-function manages a distinct segment of the journey, consistent with the rule of charging \)0.50 per each fifth of a mile or its fraction. This structure enables easy analysis of the fare charged over different mileages.
Understanding piecewise functions is crucial, as they appear in situations where different rules or formulas apply based on conditions, just like in progressive taxi fare systems.